Ordinary deformations are unobstructed in the cyclotomic limit

The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if this ring in Noetherian (a natural assumption considered by Hida) it is free over the ring of Witt vectors of $k$. This however imposes natural conditions on certain $\mu$-invariants.